### abstract

- Let be the variety of associative algebras over a field K and A = K 〈x1,…, xn〉 be a free associative algebra in the variety freely generated by a set X = {x1,…, xn}, End A the semigroup of endomorphisms of A, and Aut End A the group of automorphisms of the semigroup End A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A. A similar result is obtained for the automorphism group Aut , where is the subcategory of finitely generated free algebras of the variety . The later result solves Problem 3.9 formulated in [17].